Nicky Wire, bass player with the Manic Street Preachers, is standing on a table at Cardiff's Centre for Visual Arts. At least, a Lego model of him is. Nearby, works by Munch and Bacon rub shoulders with posters for the Sex Pistols and photos from the Vietnam war. Petitions are being signed at a stall run by Amnesty International, and Arthur Scargill is preparing to give a talk. This isn't the sort of thing you expect to find in an art gallery; no wonder the show is called Unconvention.

Galleries are increasingly looking for ways to describe art history differently, showing works thematically or comparatively, rather than grouping them by period. But what happens if you introduce a totally unexpected ordering principle - in this case, collecting art that has interested the Manic Street Preachers? Something wonderful, if it's done with the imagination and sensitivity brought to this show by its curator, the artist Jeremy Deller.

Unconvention is drawing huge crowds, and rightly so. It works because it looks right through the Manics at a radical history of modern art. From Picasso to Pollock, Unconvention implies, the great modern painters were part of a movement to change the world. Whether the Manics ever meant to suggest that all great modern art is revolutionary is beside the point. The gallery has wisely avoided consulting them.

One recognisable Manics reference is Jenny Saville's painting from the cover of their album The Holy Bible. Another, because its message was the title of one of their singles, is a Republican poster produced during the Spanish civil war. Beneath a photograph of a child killed by German bombers in league with Franco's fascists is the warning: If you tolerate this, your children will be next.

The exhibition uses this as a cue to take us into the memory of the civil war. There are Republican posters, Robert Capa's photograph of a Republican soldier being killed and, in a glass case, archives of Welsh volunteers who fought and died in Spain in the International Brigades. "Thomas Howell Jones, Working-Class Hero", reads a funeral tribute, "who died fighting Fascism at Gandesa, Spain. August 25th 1938." You turn from this and clock Picasso's Reclining Nude with Necklace (1968), lent by the Tate, and deliberately almost hidden in a corner.

Picasso was fighting the same war as Thomas Howell Jones when he painted Guernica in 1938. You suddenly get a sense of Picasso as he saw himself: a member of the Communist Party (like Howell Jones) and a revolutionary. At the same time, you look back at the funeral pamphlets and wonder if a painting is as important as someone's life.

On the other side of the room, Jackson Pollock is painting, standing in his paint-spattered boots outside his Long Island studio in Hans Namuth's famous film, showing on a video monitor. He's smoking furiously, starting to flick paint on to a canvas on the ground. His action somehow seems related to Spain too. An attempt to act in history. Another glass case holds publications by the 60s radicals the Situationists. Collages by painter Asger Jorn turn Pollock's spattery marks into a revolutionary gesture - an attack on the sameness and boredom of the social order.

This exhibition does the same thing. Looming over it all is a huge red-on-black self-portrait by Andy Warhol. Underneath, the Amnesty stall set up for the opening invites signatures for a petition against the death penalty. Bzzz - something fires in your brain as you recognise another uncanny connection: Warhol's paintings of the electric chair.

Travel and war are images that run through the exhibition. There's a Guardian page commemorating photographer Kevin Carter, who took searing shots in Africa and later killed himself (and then became the subject of a Manics song). The Situationists despised the "spectacle" of tourism, and German painter Martin Kippenberger's cynical, violent work Vom Einfachsten nach Hause depicts cheap holidays in other peoples' misery. Don McCullin's shocking black-and-white pictures of the Vietnam war are juxtaposed with official shots commissioned by the US army. Remember the TV images of the chaos at the US embassy as people scrambled on helicopters to get out of Saigon? In the army's official photograph soldiers form an orderly queue.

This is an exhibition that will leave everyone with powerful images and exhilarating thoughts. Here's one for the festive season, told in a display of Situationist literature. In the 70s, the British group King Mob arranged for a man dressed as Santa Claus to walk into Selfridges at the height of Christmas shopping to give toys to children. There was total panic. Not only was Santa dragged away by the police, but staff and constables ran around desperately taking back the toys. That's how slight and perfect a gesture it takes to destroy reality as we know it.

There are lots of stories being told in Unconvention. There's a history of pop's involvement in art. There's an eye-opening restoration of modern art into the social and political history of the 20th century. We're so used to seeing paintings by the modern masters in pristine galleries. What happens when you look at Picasso next to mementoes of the Spanish civil war, and Bacon beside photos of Vietnam? You see the grain of the painting differently. It shares its life with the world around it.

It's astonishing that as we leave this century we don't want to talk about it; we would rather remember a spurious historical epoch - the millennium. The 20th century is so full of agonising loose ends, failed radicalisms, discarded utopias, that it's easier to think about Copernicus and William the Conqueror. But this is one exhibition that does some justice to this century's loose ends and heroes.

 Unconvention is at the Centre for Visual Arts, Cardiff (02920 394040), till January 16.